How do help “Multigrid Principles based on Numerical Solutions of Partial Differential Equations” for Smoothing Process (Concepts)?

In Twenty century, Partial Differential Equations (PDE) are solved by Numerical Approach such as Adam – Smith Methods, Numerical Simulation Methods, Finite Difference methods etc., In this article we discuss about difficulties of solve either Differential equations or PDE in 3 – Dimensional cases. So that we discuss only 1 – Dimensional Multigrid Methods (M. M) and we discuss about M. M. Errors, Corrections, Type of grid such as Coarse Grid (C. G) and Fine Grid (F. G), Smoothing, non – smoothing approximation of C. G. Finally, we explain that M. G. M. works by decomposing problem into separate length scale and also using an iterative method. This method optimizes errors deduction in the length scales globally. In Multigrid Methods (MgM) several sub – routines must be developed to pass the data from C. G to F. G (Interpolation) from F. G to C. G (Reduction) and correction of the error at each grid interval (Smoothing), simply we have results reaction as (Reduction)C.G ⇄ F.G (Interpolation).


Introduction
We know that Partial Differential Equations (PDE) can be solved by analytically and numerically. Most of the PDE can be analyzed with numerical approach, for example Elliptic and Hyperbolic PDE solved by Gal grin's Method and other methods also (Hemker, 1990). In this article we develop the application of Multi -Grid Methods ( ) with the support of Functional Analysis concepts (Hackbusch, 1977). The ( ) was firstly, introduced by Hack Busch (1977) who used reliable methods. Fredrickson (1974) developed a powerful ( ) Algorithm for the Poisson Equation.
Since, the equation (1) can be described with finite differences concepts as For equation (1), the solution is denoted by and the solution of equation (3) by and its value of in by and then we conclude that is near to the solution in the grid vertex . That is why, equation (3) is called a Vertex Centered Discretization (VCD) And also assume that without loss of generality in the grid the number of members is even. We can write the equation (3) in the Matrix -Vector Form such as ℎ , the coefficient matrix , it is large, sparse and nonzero matrix.

Now, Equation (4) can be solved by Gauss -Seidal Iteration Method.
Let us take, the initial stage as 0 of equation (4) and it can be given by To identify the convergence of the solution, we will use Fourier Concepts. In this situation, we take periodic boundary conditions such as This above type of periodic grid function (7), which also be mentioned by the Fourier Series form such as Now, we can find the error that is,  (8), finally we get Note that, in equation (10), the function ( ) it is called the Amplification Factor (A. F.). This A. F. will help to measures Growth or Decay of Fourier Mode of the error during an iteration (Kettler, 1982). By some useful simple calculations, we will reach Fourier Mode Error (F. M. E.) as Again, note that, by the periodic boundary condition of the solution of equation (11) is obtained Purely Constant (Adams, 1989). Therefore, we simply ignore Fourier Mode as zero without loss of generality, decays during iterations ⟹Equation (11) is not correct measure of convergence, we can find the value as The rate of convergence as depend upon ℎ ⟶ 0 Sometime, it is true for all elliptic equations except special cases, so that, it is called Basic Iterations method (BIM)

The Important Needful of Multigrid Concepts
BIM of convergence will be improved with Multigrid Methods.
Since | ( )| decreasing (↓) as increasing (↑), so that, we got the following results a. For Long Wave Length, F. M of near to 1, that is, delay slowly and then we get b. For Short Wave Length F. M of reduced rapidly.
From those above two results, the importance of Multigrid Principle is to be approximate the smooth for the result (a), part of the error on Courser Grid (Bank & Sherman, 1981). The non smooth or rough part is reduced with small number free from the value of ℎ of iteration with a BIM on the fine Grid.
For further Multigrid procedure, we need some useful definitions

Rough Wave Number Set
It is denoted by and it is defined by In Basic Iterative Method (B. I. M) < 1, that is, it is bounded away from 1, and uniformly in ℎ, so that, this method is called Smoother and also depends on Iterative method and our problem. For Gauss -Seidel Iteration and present our model the value of it is easily calculated. Due to equation (10), | ( )| decreases monotonically and also, we get after few calculations Secondly, the Fourier Mode that cannot be represented on the Coarse Grid, which needs to reduce by B. I. M. That is why, for Coarse Grid, we simply doubling the mesh -size ℎ of . That is, due to equation (8), simply change 2 , , for our convince. Then, the balanced wave numbers are defined as Non -Smooth and also, we got by using equation (13) Equation (17) is the Smoothing Factor for Gauss -Seidal Iteration. These concepts of Gauss -Seidal Iteration, first introduced by Brandt (1977).
Suppose that, the given Smoothing Factor , it is not periodic and then are qualitatively correct but, only not correct for the case of Singular Perturbation Model.
Thirdly, with variable coefficients, a smoother factor acting very well, that is, the performs very well in the "Frozen Coefficient" case and also act very well for variable coefficients also. That is, in the frozen coefficient case the constant , as a set of constant coefficients with coefficients same as the values of the variable coefficients under enough large sample points in the domain (0, 1)

Two Grid Procedure
For this case, the smoothing part of the error can be reduced by the C. Gs, it is enough to learn the Two Grid method for our model problem.
Before create this C. G. Model Approximation, we need the following definitions, that is,

Vertex -Centered Coarsening (Definition)
Consider a Course Grid denoted by and defined by = { ∈ ℝ: = = ℎ, = 1, 2, … ℎ = 1 } Note that, all vertices of ∈ , that is why, it is called Vertex -Centered Coarsening. And also, the grid it is called

Definitions for Two Operators
Now we will introduce two operators, they are Prolongation Operator denoted by : ⟶ ℝ, It is defined by Linear Interpolation that is, All vertices, , = 1, 2, … , + 1, they are Course Grid Quantities.
Another one operator is Restriction Operator denoted by as : ⟶ is defined by ℎ , it is defined Zero, Outside that is, outside of Find Grid Set.
Form these two operators , that is, the equations (20) and (21)  The coefficient matrix is obtained by Discretizing Equation (1), so that, it is called Discretization Course Grid Approximation. Now, we can develop the Fine Gerid Problem for equation (5) by using Inner Product Approach.

Consider,
With 〈, . , 〉 the standard inner product on It is equivalent the fine grid problem in Course Grid Approximation, that is, Like Fine Grid matrix , we can find the matrix , it is Coarse Grid matrix, and it is obtained from the equation, The matrix is called Discretization Course Grid Approximation. The alternative Inner product approach, since the fine grid problem Let us take restriction of the test function to a subspace with equal dimension as , that is the functions of the type ℎ ∈ and , it is also a prolongation operator but it differs from P. Now we get Equivalent to, otherwise we have Note that * , it is either adjoint of P or transpose of P We know that, the inner product 〈, . , 〉 it is over From equation (25) we conclude that And also, we have, = ℎ = *

Remark:
The matrix , it is called Galerkin Course Grid Approximations.

Result -1: Galerkin Matrix it is equivalent to Discretization Course Grid Approximation
By using matrices , , for the matrix we will get the following results Note that, The Galerkin matrix it is equivalent to the lefthand side of discretized with Finite -Difference, so that we conclude that Galerkin Matrix ≈ Discretization Course Grid Approximation

Result -2: To Calculate Course Grid Correction
Let us take, ̃ , it is an approximation, solution of = Therefore, the error denoted by , it occurs the difference between ̃ and ⟹ ̃~= Equation (29) is approximated on Course Grid and therefore we have

To Develop Two -Grid Algorithm
First, we assume that the initial stage is 1 0 Then the first iteration starts such as , ℎ , until we will reach 0 = 1 , = 2, 3, … , That is, the initial stage is equal to first iteration and then stop iterations, otherwise continue iterations until we will reach 0 = , = 2, 3, … And note that, the number of twogrid iterations ( . ) carried out stands for 1 , this results as 1 , smoothing iterations. For example, take the previous Gauss -Seidal -Method, and then apply to = Start with initial stage as 0 , and then the first application of , it is called Pre-Smoothing and also the second application is called Post -Smoothing.
Analyzing Two -Grid Concepts 1. Clearly, the rate of Converges of Two -Grid Methods (Multi -Grid Method) is free from the meshsize that is, ℎ 2. We can analyze for 1 = 0 (No -Pre -Smooth), by using.

We will calculate, Coarse Grid Correction
Since, the error occurs by using (30) [Two -Grid -Algorithm], we get,

Verifying Smoothing
Consider the second application of , that is, post-Smoothing by Gauss -Sedal -Iterations (with initial stage as 0 ), in this case, we get the error after postsmoothing as